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The Infinite Book: A Short Guide to the Boundless, Timeless and Endless

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When we get the capacity to look closer and closer into molecules, atoms, and subatomic particles, will we always be able to find something smaller? When our telescopes or probes look deeper into space, will we always find something larger? Is there a limit to the shortness of an instant, or the duration of eternity? In _The Infinite Book: A Short Guide to the Boundless, Timeless, and Endless_, John D. Barrow has invited us to look at infinities in many ways. He's competent to do so, as a professor of mathematical sciences at Cambridge, and as the author of previous books which successfully explained such concepts as nothing or impossibility. Like his previous efforts, this is highly readable stuff, but extraordinarily mysterious. The topic is something that everyone has pondered in some way; who has not, looking into the stars, wondered how far they go? It has a universal appeal, and a history within religion, philosophy, mathematics, and physics, all of which Barrow goes into here, in an entertaining summary. There are answers here, but plenty of mysteries.

Consider a universe that is infinite in space; this is a possibility, for no one knows that space is not infinite. In a universe of infinite size anything that can happen does happen, and does so infinitely often. In such a universe, not only are you here, but somewhere out there is another you doing exactly what you are doing; in fact, there are an infinite number of you. There is also somewhere out there another you who has done everything you ever did, but on the day after his sixteenth birthday, wore black socks instead of brown. This has to be the case in an infinite universe, Barrow shows; it is enough to make us uncomfortable, but discomfort is not an argument that an infinite universe cannot exist. Mathematicians have had fun and frustration with infinity ever since Zeno, who "proved" that because to walk a mile, you first had to walk a half mile, and then a quarter mile more, and an eighth mile more after that, and so on forever, that you never were able to finish that mile, and if you are under the impression that you have accomplished a mile journey at some time, you are just deluded. The world as it perceived by us, with all its journeys, is an illusion. Georg Cantor solved the problems of dealing with infinities mathematically, but his work was viciously attacked and blocked from publication, but surprisingly, Catholic theologians welcomed his ideas as a way of understanding the infinite, the infinite that included God, of course. Today, mathematicians take Cantor's work for granted, and its religious implications are not the common stuff of sermons.

It is a pleasure to puzzle through these matters with Barrow as a guide, at least partially because this is a general overview which skims through details in order to provide a larger picture. String theory, for instance, takes a couple of pages, and cosmology not much longer. If one flavor of infinity is just too much for you to consider, another will soon present itself. Thus Barrow is able to give an accessible guide to such mathematical chestnuts as the Hotel Infinity, which although it has an infinite number of rooms and they are all occupied, can take on new guests, even an infinite number of guests, even if it has to take occupants when an infinite number of the other inns in the Hotel Infinity chain are closed. There is an examination of why the sky is dark at night, if the stars are infinite in number and there must be one out there no matter where you look. Barrow demonstrates that non-zero interest rates are evidence that time travel does not happen. He speculates about computers which could do an infinite number of tasks in a finite time; such computers might exist, and might be able to calculate the infinite digits of pi - but then how would you print it out? He shows that although we think of infinity sometimes as the biggest of numbers, infinity is not a big number, but something entirely different, and no big number ever provided such a degree of interest and research in so many fields. And as in so many of his discussions of aspects of the infinite, paradox always holds sway. We are still at the beginning of trying to find the answers to much of the material here. After all, as Barrow points out, "You can discover whether the Universe is infinite, but the learning will take an infinite time."

The value of "The Infinite Book" is definitely summed up in the chapter "The Madness of Georg Cantor." Believe it or not, "new math", that strange evolution of math teaching that stumped homework for a generation in the 60's was a direct result of Cantor's theories about sets, and the supposition that some infinite sets could be larger than others--which is the first thing you REALLY learn about infinity in mathematics.

The other great part of this book is the coupling of mathematics theory with physics. The assertion by Einstein that a singularity would be a breakdown of the laws of physics, and that any theory involving singularities would thereby have in it, the "seeds of its own destruction." Then author Barrow moves on to a very good explanation of string theory (imagine a particle that stretches like a tube in a warm enviroment, but contracts to a single point in a cool environment.) The explanations, illustrations are so clearly written in this book. It's a valuable reference for students of physics and mathematics and a great read for the curious about these subjects. Recommended.

The infinite is one of those concepts that takes just a bit of understanding. Well, maybe more than just a bit. Infinity doesn't necessarily mean a bunch, it means, mathematically, that a number cannot be determined.

Infinity is not a new idea. Mathematicians have been working on them for hundreds of years. Physicists really got involved when Einstein published the Theory of Relativity in 1905. He talked about all kinds of things happening when you approached the speed of light. Then when you actually got to the speed of light his equations went to infinity. This was a bit disconcerting. One of the real reasons for the willingness of physicists to believe in string theory is that that the equations still show valid values at and above the speed of light.

But enough on infinity. If you want to know more, here's the book for you. It discusses just about everything there is to know about infinity. It would be great for the high school math/physics teacher to use for examples. Or, it's just plain fun reading to anyone that's interested.

If you are interested in infinity and you are not familar with Cantor or Borges' "The Library of Babel", then you may be amazed by this book. Otherwise, you can find it too light. Probably good as a light summer reading.

Infinity is a fascinating subject, and I thought that this book would contain a lot of interesting information in its 300 pages. I have found many quotations, a lot of superficial theology and ethics, and little information on the concept itself. I missed more depth in handling the mathematical concepts.

Anyway, there is a very good part of the book (from my point of view) devoted to eternal inflation and simulated universes, especially for how the theories are introduced and chained. Even if it is not strictly related to infinity, it is the best part of the book. The chapter that describes Cantor's works is worth reading too.

I have not been disappointed by any of John Barrow's book so far. He has a unique gift of writing with exceptional clarity about difficult topics. This is not a typical cosmology book, but large portion is devoted to beginning, shape and future of The Universe.
Like in his previous "Book of Nothing", author mixes philosophical and scientific musings about infinities (big and small) affecting theology, mathematics, cosmology, physics (TOE) and our existence.
I found Georg Cantor's life and his quest for understanding "absolute infinity" (God?) quite interesting and emotional. And check how Blaise Pascal argued about believing (or not) in God, because of infinite gain (or loss!!).
One truth emanates from "The Infinite Book": we are far, infinitely far from knowing the truth about everything (Immanuel Kant's rings the bell!). The more we learn the bigger infinite number of questions surface in front of us. Are we nearing the limits of knowledge? Professor John Barrow does not suggest it has come to this, but read about them and enjoy stretching your mind.

This book discusses infinity. This concept has a precise definition in mathematics and since the times of Cantor we know that there are various degrees of infinity, one of the most interesting problems being whether there in an infinite between the cardinal of the natural numbers and that of the real numbers, the so called continuum hypothesis, which was proven to be undecidable in the usual Zermelo-Frankel-Choice axioms of set theory.
In recent times, cosmologists, whether those adopting the inflationary scenario or those favouring the cyclic universe, are pondering whether the universe is infinite in space and possibly eternal in time (although some believe it had a beginning about 14 billion years ago, but may never end).
So the topic of the book is pertinent to our age.
Naturally, the idea of infinite is also related to the idea of God, although this is not a scientific subject, but possibly a philosophical one.
The first part of the book is a hystorical review of the concept of infinity, from Zeno and Aristotle to Kant and Cantor, via St. Augustine. A very entertaining chapter is the one about the Hotel Infinity and all the challenges that the manager meets, quite successfully and that would be impossible in a hotel with only a finite number of rooms. The second part of the book deals more with physics and cosmology, things like the singularities at the center of black holes. It is interesting to learn that an English astronomer of the 16th century already proposed that the universe is infinite. The question of the possible topologies of the universe is discussed, although we do not know yet the answer. The important distinction between the observable universe and the universe as such is made in page 139 where the radius of the visible universe is stated to be 42 billion light years (which seems to be the correct figure if we take into account the expansion of the universe since the light emitted 14 billion years ago has reached us). Unhappily , the drawing in the next page will confound the lay reader because the radius is pictured at 14 billion light years. (There are also some other minor mistakes in the book, which would have been avoided by a careful reviewer before publishing. Another example is the graph in page 190 which suggests that expansion of the universe is decelerating, contrary to recent data of supernovas). Naturally, the limit on how fast information can spread will probably preclude us from knowing whether the universe is infinite unless we can get some degree of confidence on some basic theory that predicts this infinity.
The book also discusses interesting problems regarding the impact on ethics of inmortality and the possibility of clones in an infinite universe (Vilenkin has explored also this idea in one of his books). Physicists have changed their views on the universe in the last 30 years when it was hoped that The Theory of Everything would be mathematically unique and would determine one universe. Instead, superstring theory has landed with a whole landscape of possible universes. So the question remains, how we happen to live in such universe that has made it possible for life to appear (at least in the Earth, possibly in many other planets) and to develop a self-conscious and inquisitive species by means of which the universe interrogates itself? The diverse answers are tabulated in page 186.
It also has another chapter on virtual reality "à la Matrix" (simulated universes) and it also discusses the possibility that advanced civilizations are capable of cultivating universes, the way we grow cornfields or build cities.
Another of the subjects discussed by the author is that of machines capable of supertasks . I found very interesting the 4-body configuration discovered by Xia in 1971 that , according to Newton's theory , sends the 4 bodies at infinite distance in finite time. Einstein's general relativity doesn't allow this, so that infinities did appear not only in quantum mechanics, but also in newtonian mechanics.
One of the important conclusions of the book is that the human race is not necessarily equipped to know all things that are true about the universe. "We have no special right to expect that all truths about the Universe can be tested by observations that are within our reach: that really would be an anti-Copernican outlook" (page 198).
The book is an eye opener for those readers not familiar with the role of infinity in the mathematical and physical sciences, but if you look for definite answers about these difficult problems you will not find them here (not in other books, of course).

As I started studying Calculus more and more it made me a lot more curious about the nature of infinity. We take these limits of functions to get the derivation process, we look at area with integration by summing infinitely small pieces under a curve, and then we look at the divergence and convergence of a series with limits to see what these things do at infinity. Then, to top it off, I had my mind further blown by discussing infinite dimensions in Linear Algebra. Needless to say, I had infinity on my mind. However, I wanted a more informal book to read to pass the time. Studying the formal texts is fun, but can get tedious.

Barrow's book "The Infinite Book" is exactly what I was looking for. Don't worry, if you're not a mathematician you can still grasp the general concepts discussed in this book. One of the great aspects of this book is that Barrow brings through a bit of the history of infinity. With that in mind you take a walk through the history of infinity and how it touches human curiosity. Natural he makes reference to Zeno's Paradox. (I'm actually getting sick of reading about this, it feels like just about every math book I read discusses Zeno. Don't get me wrong, it's definitely relevant.) Anyway, some of the more interesting stuff comes when he gets to discussing Cantor's history. I found this utterly fascinating and Cantor did some impressive work with infinity.

Naturally Barrow ups the ante as the history gets to more modern day applications. Cosmology being a major one, but he also brings us through some of the more "pop culture" elements (if I dare call them that). Such as strange movies that involve things like an infinite game, which are now movies I intend to track down and watch. Some of this material may be beyond some readers, it was a little beyond me at times, but I wouldn't say it was so overwhelming I got totally lost in what was being discussed.

If you're at all curious about the nature of infinity, then I highly recommend this book. I would even rank this as a rather exhilarating read. Barrow approaches the subject in an excellent way and not in a way where I think non-mathematicians will feel put aback by the arcane discussions. After reading this I am definitely interested in reading other books by Barrow and I would definitely recommend this to anyone interested.

Barrow is one of my favorite authors, and his lucid explanation of fascinating topics is always a pleasure.

The Book starts off pretty well, very interesting, brings up unique stories, concept and theories on infinity. Half way through the Book, everything is already said, and begins to repeat itself (like infinity!). The book speaks too often about the universe and its implication in regards to infinity. In conclusion starts good and then drops some, but very thought provoking, and enjoyable.

A superb book.Requires a certain level of understanding of physics and philosophy for access into its depth,but can also be read by a high school student.Proves the truth of the Orthodox Christian doctrine of One infinite God with Three equal members[Father,Son,Holy Spirit-Holy Triad].